Alternating Current

AS Unit 4 Option A: Alternating Current Learners should be able to demonstrate and apply their knowledge and understanding of:
a)	Using Faraday’s law, the principle of electromagnetic induction applied to a rotating coil in a magnetic field
b)	The idea that the flux linkage of a rotating flat coil in a uniform magnetic B-field is [math]BAN cos ωt[/math] because the angle between the coil normal and the field can be expressed as [math]θ = ωt[/math]
c)	The equation [math]V = ωBANsin ωt[/math] for the induced emf in a rotating flat coil in a uniform B-field
d)	The terms frequency, period, peak value and rms value when applied to alternating potential differences and currents
e)	The idea that the rms value is related to the energy dissipated per cycle, and use the relationships [math]I = \frac{I_0}{\sqrt{2}} \text{ and } \quad V = \frac{V_0}{\sqrt{2}} (\text{ including }V_{\text{rms}} = \frac{\omega B A N}{\sqrt{2}})[/math]
f)	The idea that the mean power dissipated in a resistor is given by [math]P = IV = I^2 R = \frac{V^2}{R}[/math] where V and I are the rms values
g)	The use of an oscilloscope (CRO or PC based via USB or sound card) to measure • a.c. and d.c. voltages and currents • frequencies
h)	The 90° phase lag of current behind potential difference for an inductor in a sinusoidal a.c. circuit
i)	The idea that [math]X_L = \frac{V_{\text{rms}}}{I_{\text{rms}}} [/math] is called the reactance, [math]X_L[/math], of the inductor, and to use the equation [math]X_L = ωL[/math]
j)	The 90° phase lead of current ahead of potential difference for a capacitor in a sinusoidal a.c. circuit, and to use the equation [math]X_C = \frac{V_{\text{rms}}}{I_{\text{rms}}}, \quad \text{where} \quad X_C = \frac{1}{\omega C}[/math]
k)	The idea that the mean power dissipation in an inductor or a capacitor is zero
l)	How to add potential differences across series RC, RL and RCL combinations using phasors
m)	How to calculate phase angle and impedance, Z, (defined as [math]Z = \frac{V_{\text{rms}}}{I_{\text{rms}}}[/math] for such circuits)
n)	How to derive an expression for the resonance frequency of an RCL series circuit
o)	The idea that the Q factor of a RCL circuit is the ratio [math]\frac{V_L}{V_R} = (\frac{V_C}{V_R})[/math] at resonance
p)	The idea that the sharpness of the resonance curve is determined by the Q factor of the circuit

a) Electromagnetic Induction in a Rotating Coil
⇒ Faraday’s Law
Faraday’s law of electromagnetic induction states that the induced electromotive force (emf) in a coil is equal to the negative rate of change of the magnetic flux linkage through the coil. Mathematically, for a coil with N turns:
[math]\varepsilon = -\frac{d\Phi_{\text{total}}}{dt}[/math]
Where:
– ε is the induced emf,
– [math]\Phi_{\text{total}}[/math] = NΦ is the total flux linkage,
– Φ is the magnetic flux through one turn of the coil.
Figure 1 Faraday’s Law of Electromagnetic induction
b) Flux Linkage for a Rotating Flat Coil
Consider a flat coil with:
– N turns,
– Each turn having an area A,
– Immersed in a uniform magnetic field B.
If the coil rotates at an angular velocity ω (radians per second), the angle θ between the normal to the coil and the magnetic field changes with time as:
[math]θ = ωt[/math]
The magnetic flux through one turn is given by:
[math]Φ = BAcosθ = BAcos(ωt)[/math]
Thus, the total flux linkage is:
[math]Φ_{total} = NΦ = BANcos(ωt)[/math]
Figure 2 Flux linkage for a rotating flat coil
⇒ Derivation of the Induced emf
Using Faraday’s law, the induced emf is:
[math]\varepsilon = \frac{d}{dt} \left(-BAN \cos(\omega t) \right) \\
\varepsilon = -BAN \frac{d}{dt} \left(\cos(\omega t) \right)[/math]
Since B, A, and N are constants, differentiating with respect to time gives:
[math]ε = -BAN(-ωsin (ωt))[/math]
Thus:
[math]ε = BANωsin (ωt))[/math]
This equation shows that the induced emf is sinusoidal with an angular frequency ω, a peak (maximum) value of [math]ωBAN[/math], and a phase shift determined by the sine function.
c) Definitions of Frequency, Period, Peak Value, and RMS Value
⇒ Frequency and Period
Frequency (f) is the number of cycles of the oscillatory motion per unit time. It is measured in hertz (Hz).
Angular Frequency (ω) is related to the frequency by:
[math]ω = 2πf[/math]
Period (T) is the time taken for one complete cycle of the waveform. It is the reciprocal of the frequency:
[math]T = \frac{1}{f} \quad \text{or} \quad T = \frac{2\pi}{\omega}[/math]
Figure 3 Difference between frequency and period
⇒ Peak Value
Peak Value (or Amplitude):
– The maximum instantaneous value of the voltage (or current) waveform.
– In our induced emf equation, the peak value is:
[math]ε_{peak} = ωBAN[/math]
⇒ RMS Value (Root Mean Square)
Definition:
– The RMS value of an alternating voltage or current is the effective value that delivers the same power as an equivalent DC source. For a sinusoidal waveform, the RMS value is given by:
[math]V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}}[/math]
Similarly, for a sinusoidal current:
[math]I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}}[/math]
⇒ RMS:
– Since alternating currents and voltages vary over time, the RMS value provides a useful measure for comparing AC with DC in terms of power delivery.
Figure 4 RMS of an alternating current
d) Graphical Representation
Displacement of Flux and Induced emf
⇒ Flux Linkage Graph:
If you plot the magnetic flux linkage [math]Φ_{total} = BANcos(ωt)[/math] versus time, you obtain a cosine wave oscillating between +BAN and −BAN.
⇒ Induced emf Graph:
The induced emf [math]ε = ωBANsin(ωt)[/math] is a sine wave. Note the 90° phase difference (the sine wave is shifted by [math]\frac{π}{2}[/math] relative to the cosine wave of flux). The maximum (peak) emf is [math]ωBAN[/math]
⇒ Waveform Characteristics:
Frequency (f) determines how quickly the waveform repeats.
Period (T) is the duration of one complete cycle.
Peak Value is the maximum amplitude reached.
RMS Value is the effective value for power calculations, which for a sine wave is [math]\varepsilon_{\text{rms}} = \frac{\varepsilon_{\text{peak}}}{\sqrt{2}}[/math]
e) RMS Value and Energy Dissipation
⇒ RMS Value Fundamentals
For a sinusoidal waveform, the instantaneous voltage (or current) varies with time. The peak value ( [math]V_o[/math] or [math]I_o[/math]) is the maximum amplitude reached. However, because power dissipation depends on the square of the voltage or current, the effective value is given by the root-mean-square (rms) value. For a pure sine wave:
[math]V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \\
I_{\text{rms}} = \frac{I_0}{\sqrt{2}}[/math]
In a rotating coil (as in a generator) with induced emf given by:
[math]V(t) = ωBANsin(ωt)[/math]
The peak voltage is [math]ωBAN[/math] Thus, the rms value of the induced voltage is:
[math]V_{\text{rms}} = \frac{\omega B A N}{\sqrt{2}}[/math]
⇒ Energy Dissipation Connection
The energy dissipated in a resistor over one complete cycle of an AC waveform is related to the square of the rms value. This is because the instantaneous power P(t) is:
[math]P(t) = \frac{v^2(t)}{R} \\
P(t) = i^2(t) R[/math]
When you average this power over a complete cycle, you obtain:
[math]\langle P \rangle = \frac{V_{\text{rms}}^2}{R} \\
\langle P \rangle = I_{\text{rms}}^2 R \\
\langle P \rangle = V_{\text{rms}} I_{\text{rms}}[/math]
Thus, the rms values directly reflect the effective energy (and power) delivered by an AC source.
f) Mean Power Dissipated in a Resistor
For a resistor subjected to an AC voltage and current (both expressed in rms values), the mean power is given by:
[math]P = V_{\text{rms}} I_{\text{rms}} \\
P = I_{\text{rms}}^2 R \\
P = \frac{V_{\text{rms}}^2}{R}[/math]
This expression is crucial because it tells us that even though the instantaneous voltage and current vary sinusoidally, the effective (average) power can be calculated using the rms values.
Figure 5 Mean power dissipated in a resistor
g) Using an Oscilloscope for Measurements
⇒ Measurement Capabilities:
AC and DC Voltages/Currents:
Modern oscilloscopes (whether traditional Cathode Ray Oscilloscopes, CROs, or PC-based via USB/sound card) allow you to visualize waveforms. You can measure:
– DC voltage levels: by observing the baseline.
– AC voltages: by displaying the time-varying waveform.
– Current: typically measured indirectly using a current probe or a shunt resistor (where the voltage drop across the resistor is proportional to the current).
Figure 6 Voltage and current measurement with an Oscilloscope
Frequency Measurement:
The oscilloscope can display the period T of the waveform directly, from which the frequency
[math]f = \frac{1}{T} \quad \text{or} \quad f = \frac{\omega}{2\pi}[/math]
is calculated.
⇒ Practical Steps:
1. Connect the Signal:
– Attach the probe to the circuit (ensuring proper grounding).
2. Set Time Base and Voltage Scale:
– Adjust the time and voltage scales so the waveform is clearly visible.
3. Measure Peak and RMS Values:
– Some modern scopes automatically compute rms values.
– Alternatively, use the waveform statistics or manually determine the peak and compute rms as [math]V_{\text{rms}} = \frac{V_0}{\sqrt{2}}[/math] for a sinusoid.
4. Frequency Determination:
– Measure the period on the time axis and calculate frequency as
[math]f = \frac{1}{T}[/math]
h) Phase Relationships in Inductive AC Circuits
⇒ Inductive Behavior and Phase Lag:
For an inductor in a sinusoidal AC circuit, the voltage across the inductor leads the current by 90° (or the current lags behind the voltage by 90°). This occurs because:
– The voltage across an inductor is given by:
[math]v_L(t) = L \frac{di}{dt}[/math]
– For a current [math]i(t) = I_0 sin(ωt)[/math] the derivative is:
[math]\frac{di}{dt} = \frac{d}{dt} I_0 \sin(\omega t) \\
\frac{di}{dt} = I_0 \omega \cos(\omega t) \\
\frac{di}{dt} = I_0 \omega \sin\left(\omega t + \frac{\pi}{2}\right)[/math]
Hence, the induced voltage is:
[math]v_L(t) = L I_0 \omega \sin\left(\omega t + \frac{\pi}{2}\right)[/math]
This shows the voltage leads the current by 90°.
⇒ Graphical Representation:
On an oscilloscope, if you display both the current and the voltage waveforms:
– The voltage waveform (across the inductor) will reach its maximum [math]π/2[/math] (90°) before the current waveform.
– This phase shift is a key signature of inductive behavior in AC circuits.
Figure 7 Inductance in AC Circuits
i) Inductive Reactance, [math]X_L[/math]
⇒ Definition and Equation:
The reactance of an inductor, X_L, quantifies the opposition that an inductor presents to a changing current. It is defined as:
[math]X_L = \frac{V_{\text{rms}}}{I_{\text{rms}}}[/math]
for an inductor in a sinusoidal AC circuit.
The theoretical relationship between the reactance and the inductor’s inductance L is given by:
[math]X_L = ωL[/math]
Where:
– ω is the angular frequency ( [math]ω = 2πf[/math]),
– L is the inductance in Henries (H).
⇒ Phase Relationship:
In an inductor, the induced emf (voltage) is proportional to the time derivative of the current:
[math]v_L(t) = L \frac{di}{dt}[/math]
For a sinusoidal current,[math]i(t) = I_0 sin(ωt)[/math] , the derivative is:
[math]\frac{di}{dt} = I_0 \cos(\omega t) \\
\frac{di}{dt} = I_0 \omega \sin\left(\omega t + \frac{\pi}{2}\right)[/math]
Thus, the voltage across the inductor is:
[math]v_L(t) = L I_0 \omega \sin\left(\omega t + \frac{\pi}{2}\right)[/math]
This shows that the voltage (and hence the effective [math]V_{rms}[/math]) leads the current by 90°. Equivalently, one says that the current lags the voltage by 90° in an inductor.
j) Capacitive Reactance,[math]X_C [/math
⇒ Definition and Equation:
The reactance of a capacitor,[math]X_C [/math] , represents the opposition a capacitor offers to a change in voltage. It is defined as:
[math]X_C = \frac{V_{\text{rms}}}{I_{\text{rms}}}[/math]
For a capacitor with capacitance C, the capacitive reactance is given by:
[math]X_C = \frac{1}{\omega C}[/math]
Where:
– ω is the angular frequency,
– C is the capacitance in farads (F).
Figure 8 Capacitance reactance
⇒ Phase Relationship:
The current through a capacitor is given by:
[math]i_C(t) = C \frac{dv}{dt}[/math]
For a sinusoidal voltage,[math]v(t) = V_0 \sin(\omega t)[/math], the derivative is:
[math]\frac{dv}{dt} = \frac{d}{dt} \left( V_0 \sin(\omega t) \right) \\
\frac{dv}{dt} = V_0 \omega \cos(\omega t) \\
\frac{dv}{dt} = V_0 \omega \sin(\omega t + \frac{\pi}{2})[/math]
Thus, the current becomes:
[math]i_C(t) = C V_0 \omega \sin(\omega t + \frac{\pi}{2})[/math]
This shows that the current leads the voltage by 90° in a capacitor.
k) Zero Mean Power Dissipation in Pure Reactances
For both pure inductors and pure capacitors, the instantaneous power is given by:
[math]P_(t) = v(t) \cdot i(t)[/math]
Because of the 90° phase difference (either voltage leading or lagging the current), the instantaneous product is a sinusoidal function that averages to zero over one complete cycle.
Mathematically, for a sinusoidal waveform:
[math]P_{\text{mean}} = \frac{1}{T} \int_0^T v(t) \cdot i(t) \, dt \\
P_{\text{mean}} = 0[/math]
Therefore, in ideal (lossless) inductors and capacitors, no net (mean) power is dissipated; the energy is temporarily stored and then returned to the circuit.
l) Phasor Addition in Series RC, RL, and RCL Circuits
⇒ Phasor Concept:
Phasors are a way to represent sinusoidally varying quantities (such as voltage and current) as complex numbers or rotating vectors.
They allow for the algebraic addition of AC voltages and currents, taking into account both their magnitudes and phase angles.
⇒ Example 1: Series RC Circuit
Components:
– Resistor R (voltage in-phase with current).
– Capacitor C (voltage lags current by 90°).
Phasor Representation:
– Let [math]V_R[/math] be the voltage across the resistor and [math]V_C[/math] the voltage across the capacitor.
– The total voltage is:
[math]V_{\text{total}} = V_R + V_C[/math]
⇒ Addition:
Since [math]V_R[/math] and [math]V_C[/math] are 90° apart, their phasor addition is done by vector addition in the complex plane:
[math]V_{\text{total}} = \sqrt{(V_R)^2 + (V_C)^2}[/math]
The phase angle of [math]V_{\text{total}}[/math] is given by:
[math]\phi = \tan^{-1} \left( \frac{V_C}{V_R} \right)[/math]
⇒ Example 2: Series RL Circuit
Components:
– Resistor R (voltage in-phase with current).
– Inductor L (voltage leads current by 90°).
Phasor Addition:
– Let [math]V_R[/math] be the voltage across the resistor and [math]V_L[/math] the voltage across the inductor.
– The total voltage:
[math]V_{\text{total}} = \sqrt{(V_R)^2 + (V_L)^2}[/math]
– And the phase angle is:
[math]\phi = \tan^{-1} \left( \frac{V_L}{V_R} \right)[/math]
⇒ Example 3: Series RCL Circuit
Components:
– Resistor R,
– Capacitor C,
– Inductor L.
Phasor Representation:
– [math]V_R[/math] is in phase with current.
– [math]V_L[/math] leads current by 90°.
– [math]V_C[/math] lags current by 90°.
Net Reactance:
– The net reactive voltage is:
[math]V_{\text{reactive}} = V_L – V_C[/math]
Total Voltage:
– The overall voltage is the vector sum:
[math]V_{\text{total}} = \sqrt{V_R^2 + (V_L – V_C)^2}[/math]
Phase Angle:
– The phase angle is:
[math]\phi = \tan^{-1} \left( \frac{V_L – V_C}{V_R} \right)[/math]
m) Phase Angle and Impedance in an RCL Series Circuit
⇒ Impedance, Z
For an RCL series circuit, the circuit elements are:
– Resistor, R (with voltage [math]V_R[/math])
– Inductor, L with inductive reactance:
[math]X_L = \omega L[/math]
– Capacitor, CCC with capacitive reactance:
[math]X_C = \frac{1}{\omega C}[/math]
Because the resistor’s voltage is in phase with the current, while the inductor’s voltage leads the current by 90° and the capacitor’s voltage lags the current by 90°, the overall impedance is a vector sum:
[math]Z = \sqrt{R^2 + (X_L – X_C)^2} \\
Z = \sqrt{R^2 + \left(\omega L – \frac{1}{\omega C}\right)^2}[/math]
Figure 9 Series RLC circuit
⇒ Phase Angle, ϕ
The phase angle ϕ between the total voltage and the current is given by:
[math]\tan \phi = \frac{X_L – X_C}{R} \\
\tan \phi = \frac{\omega L – \frac{1}{\omega C}}{R}[/math]
– If [math]X_L > X_C[/math]:
The circuit is inductive and ϕ is positive (voltage leads current).
– If [math]X_C > X_L[/math]:
The circuit is capacitive and ϕ is negative (current leads voltage).
– At resonance:
[math]X_C = X_L, \text{ so } \tan \phi = 0 \text{ and } \phi = 0^\circ[/math](voltage and current are in phase).
n) Resonance Frequency of an RCL Series Circuit
Resonance occurs when the inductive and capacitive reactances cancel each other, i.e.,
[math]X_C = X_L \\
\omega L = \frac{1}{\omega C}[/math]
Solving for ω:
[math]\omega^2 = \frac{1}{LC} \\
\omega_0 = \frac{1}{\sqrt{LC}}[/math]
The resonance frequency [math]f_0[/math] (in Hz) is then:
[math]f_0 = \frac{\omega_0}{2\pi} \\
f_0 = \frac{1}{2\pi\sqrt{LC}}[/math]
At resonance, the impedance becomes purely resistive (Z=R) and the voltage and current are in phase.
o) Q Factor of an RCL Circuit
⇒ Definition of Q Factor
The quality factor (Q) is a measure of the “sharpness” of the resonance, which relates the energy stored to the energy dissipated per cycle. For a series RCL circuit, one common expression is:
[math]Q = \frac{1}{R} \sqrt{\frac{L}{C}}[/math]
Alternatively, at resonance, the voltage across the inductor (or capacitor) can be much larger than the applied voltage across the resistor. Hence, Q can also be defined as:
[math]Q = \frac{V_L}{V_R} \quad \text{or} \quad Q = \frac{V_C}{V_R}[/math]
⇒ Interpretation:
High Q Factor:
– A high Q indicates low damping, meaning that the circuit stores energy for many cycles before it is dissipated. The resonance curve is sharp and narrow, leading to high selectivity.
Low Q Factor:
– A low Q indicates higher damping, a broader resonance curve, and lower selectivity.
Figure 10 Q factor in electrical and electronics
p) The idea that the sharpness of the resonance curve is determined by the Q factor of the circuit
The Q factor (quality factor) of a circuit is a measure of how “selective” or “sharp” its resonant response is. In a resonant circuit (such as an RCL series circuit), the Q factor can be defined as:
[math]Q = \frac{\omega_0}{\Delta \omega}[/math]
Where:
– [math]\omega_0[/math] is the resonance (angular) frequency.
– [math]\Delta \omega[/math] is the bandwidth over which the power drops to half its maximum value.
A high Q factor means that the circuit has low damping and stores energy efficiently relative to the energy it dissipates. This results in a very narrow (sharp) resonance curve—only a small range of frequencies around [math]\omega_0[/math] will be strongly amplified. Conversely, a low Q factor implies higher damping, leading to a broader resonance curve with less selectivity.
⇒ Key Points:
1. Energy Perspective:
A high-Q circuit has a high ratio of energy stored in its reactive components (inductors and capacitors) to the energy dissipated per cycle (mainly by resistance). This means the circuit oscillates for many cycles before the energy decays significantly, resulting in a narrow frequency range of high amplitude.
2. Voltage Ratios at Resonance:
In a series RCL circuit at resonance, the voltages across the inductor and capacitor can be much larger than the voltage across the resistor (i.e. [math]\frac{V_L}{V_R} = Q \quad \text{and} \quad \frac{V_C}{V_R} = Q[/math]). This again indicates that a high-Q circuit has a pronounced resonance, where the reactive voltages peak sharply at the resonant frequency.
3. Selectivity and Bandwidth:
The sharper the resonance (higher Q), the more the circuit will “select” or favor a narrow band of frequencies. This is useful in applications like filters and tuned amplifiers where you want to isolate or amplify a specific frequency.
⇒ Graphical Illustration:
Imagine plotting the amplitude of the circuit’s response (for example, the voltage across the resistor) against frequency.
A high-Q circuit will have a very narrow, tall peak at [math]ω_0[/math]. Only frequencies very close to [math]ω_0[/math] will result in a strong response, while frequencies even slightly away from [math]\omega_0[/math] will produce much lower amplitudes.
A low-Q circuit will display a broader, lower peak, meaning the circuit responds to a wider range of frequencies, but with less selectivity.